Theorem ccatvalfn

Description: The concatenation of two words is a function over the half-open integer range having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018)

	Ref	Expression
Assertion	ccatvalfn	⊢ ( ( 𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ) → ( 𝐴 ++ 𝐵 ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ccatfval	⊢ ( ( 𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ) → ( 𝐴 ++ 𝐵 ) = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) )
2		fvex	⊢ ( 𝐴 ‘ 𝑥 ) ∈ V
3		fvex	⊢ ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ∈ V
4	2 3	ifex	⊢ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ∈ V
5		eqid	⊢ ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) )
6	4 5	fnmpti	⊢ ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) )
7		fneq1	⊢ ( ( 𝐴 ++ 𝐵 ) = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) → ( ( 𝐴 ++ 𝐵 ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↔ ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ) )
8	6 7	mpbiri	⊢ ( ( 𝐴 ++ 𝐵 ) = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↦ if ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) , ( 𝐴 ‘ 𝑥 ) , ( 𝐵 ‘ ( 𝑥 − ( ♯ ‘ 𝐴 ) ) ) ) ) → ( 𝐴 ++ 𝐵 ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) )
9	1 8	syl	⊢ ( ( 𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ) → ( 𝐴 ++ 𝐵 ) Fn ( 0 ..^ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) )